Integration Query

Say we want to find . Normally, we let , or realise that the integrand is the derivative of , or do something along similar lines, but suppose we ignore those. Instead, we do the following:

Write and let:

.

More generally, letting gives:

.

My Question (1): why are these identities not used and instead the arctangent ones are? I even suspected that I was missing something, but differentiating quickly confirmed that they are indeed correct.

We know that

But using the relation found above gives:

, which doesn't converge, since the logarithmic function is not defined on .

Question (2): why is this? I probably need to pick up a complex analysis book!

My Question (1): why are these identities not used and instead the arctangent ones are?

The obvious answer is that this approach takes a nice simple trigonometric integral and replaces it by something involving complex logarithms, which are difficult beasts to tame.

Originally Posted by TheCoffeeMachine

But using the relation found above gives:

, which doesn't converge, since the logarithmic function is not defined on .

Question (2): why is this? I probably need to pick up a complex analysis book!

A complex analysis book would be a good investment.

The reason things look wrong here is that when you make the substitution , as x goes from 0 to ∞, t goes from –1 to 1. But it does not go from –1 to 1 along the real axis, it goes (anticlockwise) round the unit circle in the complex plane. The integral of 1/t is log(t), and the values of log(t) at –1 and 1 are and . Taking the integral along that contour, you get the result , which is what you would expect from .

The obvious answer is that this approach takes a nice simple trigonometric integral and replaces it by something involving complex logarithms, which are difficult beasts to tame.

Well understood!

A complex analysis book would be a good investment.

Feel free to recommend one, please.

The reason things look wrong here is that when you make the substitution , as x goes from 0 to ∞, t goes from –1 to 1. But it does not go from –1 to 1 along the real axis, it goes (anticlockwise) round the unit circle in the complex plane. The integral of 1/t is log(t), and the values of log(t) at –1 and 1 are and . Taking the integral along that contour, you get the result , which is what you would expect from .